Software utilizing boundary integral equation techniques to solve electromagnetic scattering problems (e.g., using the method of moments) has existed at least since the 1960s. Such techniques have the potential for making very accurate radar cross-section predictions. However, practical application of the techniques to realistic problems is severely limited by poor scalability of computational resource requirements (i.e., CPU time and memory) relative to the problem size and complexity. With the relatively recent introduction of computational techniques involving high-order and “fast” mathematical methods (such as the fast multipole method (FMM)), the situation has changed drastically. Now, instead of the resource requirements for a problem described by N unknowns scaling in O(N3) time, they scale in O(NlogN) time, much like the Fast Fourier Transform (FFT) algorithm. Consequently, it has become possible to handle a much wider array of problems of practical interest.
Although the methods used have improved drastically, fast methods must still be used in conjunction with iterative solvers. If the set of linear equations that arises from discretizing a boundary integral equation is ill conditioned, the number of iterations required to obtain an accurate solution is not well controlled, and may be very large. The overall efficiency of the solution process suffers accordingly. In addition, a high degree of numerical precision is generally required in the solution process, causing the need for rounding, and resulting in a loss of solution accuracy.
Unfortunately, the most widely used integral equation formulation for closed, perfect electrical conductors (PEC), namely, the combined field integral (CFIE) equation, is not, in general, well conditioned. The ill conditioning stems from the fact that the CFIE necessarily includes an integral operator component that is “hypersingular”. This component is, in fact, not simply an integral operator at all, but rather the gradient of an integral operator. Whereas, ordinary integral operators tend to filter out the numerical “noise” associated with a discretization, differential operators tend to amplify it.
The eigenvalue spectrum of a hypersingular integral operator is unbounded, which leads directly to an ill-conditioned linear system. The ill effects become evident in the form of an increase in the required number of iterations as one refines the discretization of a target. Fine discretizations are often required to spatially resolve source variations or geometric detail on the subwavelength scale. This difficulty with the CFIE (as well as with some other boundary integral formulations in computational electromagnetics) is known as the “low frequency problem.”
Various methods have been employed to overcome this problem. The most commonly practiced method is to multiply the CFIE matrix by a “preconditioner” matrix at every step of the iterative solver procedure. An effective preconditioner improves the condition number of a linear system, thereby reducing the number of iterations required for solution. Examples of preconditioners that have been utilized are the block-diagonal preconditioner, the incomplete LU preconditioner, and the sparse approximate inverse preconditioner. Preconditioners such as these are general purpose—in other words, they can be applied with varying degrees of success to any matrix. Unfortunately, however, none of them takes specific advantage of the analytical properties of the CFIE, so none of them is ideally suited for use with the CFIE.
Instead of preconditioning the equation, it is also possible to “post-condition” it. This involves defining the unknown part of the original linear equation to be the driving term for some new linear system. If the combined linear system is properly conditioned, then the problem is effectively solved because, once the new auxiliary unknowns have been solved for, it is straightforward to determine the solution to the original linear system. The moment method approaches that use “loop-star” basis functions fall into this category. While some success has been achieved with this technique, it is cumbersome with limited extensibility, and has been applied only to low-order, moment method discretizations.
Therefore, it is desirable to provide a preconditioner method that is specifically tailored for use with the CFIE. More specifically, the preconditioner should be designed to minimize the number of iterations required for a solution. This is especially desirable when the use of a fine discretization to resolve source variations or geometric detail on a subwavelength scale results in an ill-conditioned linear equation (the “low-frequency” problem). Most specifically, it is desirable to provide a computer program product, an apparatus, and a method for modeling the electromagnetic response of a closed arbitrarily shaped three-dimensional object to an arbitrary time-harmonic incident field by means of a well-conditioned boundary integral equation (BIE) by discretizing the well-conditioned BIE to provide and solve a well-conditioned finite dimensional linear system to determine the distribution of equivalent surface sources on the object.
References
    [1] Leslie Greengard and Stephen Wandzura. Fast multipole methods. IEEE Computational Science & Engineering, 5(3):16-18, July 1998.    [2] John J. Ottusch, Mark A. Stalzer, John L. Visher, and Stephen M. Wandzura. Scalable electromagnetic scattering calculations for the SGI Origin 2000. In Proceedings SC99, Portland, Oreg., November 1999. IEEE.    [3] Jiming M. Song and Weng Cho Chew. The fast Illinois solver code: Requirements and scaling properties. IEEE Computational Science & Engineering, 5(3):19-23, July 1998.    [4] Andrew F. Peterson. The “interior resonance” problem associated with surface integral equations of electromagnetics: Numerical consequences and a survey of remedies. Electromagnetics. 10:293-312, 1990.    [5] George C. Hsiao and Ralph E. Kleinman. Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics. IEEE Transactions on Antennas and Propagation, 45(3):316-328, March 1997.    [6] Nagayoshi Morita, Nobuaki Kumagai, and Joseph R. Mautz. Integral Equation Methods for Electromagnetics. Artech House, Boston, 1990.    [7] Robert J. Adams. A Class of Robust and Efficient Iterative Methods for Wave Scattering Problems. Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Va., December 1998.    [8] Robert J. Adams and Gary S. Brown. Stabilization procedure for electric field integral equations. Electronic Letters, 35(23):2015-2016, November 1999.    [9] Peter Kolm and Vladimir Rokhlin. Quadruple and octuple layer potentials in two dimensions I: Analytical Apparatus. Technical Report YALEU/DCS/RR-1176, Yale University, Department of Computer Science, March 1999.    [10] Barrett O'Neil. Elementary Differential Geometry. Academic Press, New York, 1997.    [11] John David Jackson. Classical Electrodynamics. John Wiley & Sons, New York, second edition, 1975.    [12] Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions. Applied Mathematics Series. National Bureau of Standards, Cambridge, 1972.    [13] Lawrence S. Canino, John J Ottusch, Mark A. Stalzer, John L. Visher, and Stephen M. Wandzura. Numerical solution of the Helmholtz equation in 2d and 3d using a high-order Nyström discretization. Journal of Computational Physics, 146:627-663, 1998.